发表机构
School of Mathematics, South China University of Technology; School of Mathematics and Statistics, Guangdong University of Technology（华南理工大学数学学院; 广东工业大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明在加权Sobolev空间中，二维不可压缩Navier-Stokes方程在Poiseuille流附近的初始扰动大小为O(ν^{2/3})时，流动全局非线性稳定，非零模态以增强耗散指数衰减，零模态受控，且该稳定性尺度无对数损失。
AI 中文摘要
我们研究了在二维不可压缩Navier--Stokes方程中，二次平面Poiseuille流$U(y)=(y^2,0)$在环面$\mathbb T\times\mathbb R$上的非线性稳定性。我们证明了对于在依赖于$\nu$的加权Sobolev空间中大小为$O(\nu^{2/3})$的初始扰动，该流动是全局非线性稳定的。特别地，非零流向模态经历增强耗散，并在特征时间尺度$O(\nu^{-1/2})$上指数衰减，而零模态保持均匀受控。因此，在此加权Sobolev拓扑中，$O(\nu^{2/3})$是一个无对数损失的充分稳定性尺度。
英文摘要
We study the nonlinear stability of the quadratic plane Poiseuille flow $U(y)=(y^2,0)$ for the two-dimensional incompressible Navier--Stokes equations on $\mathbb T\times\mathbb R$. We prove that for initial perturbations of size $O(ν^{2/3})$ in a $ν$-dependent weighted Sobolev space, the flow is globally nonlinearly stable. In particular, the nonzero streamwise modes experience enhanced dissipation and decay exponentially on the characteristic time scale $O(ν^{-1/2})$, while the zero mode remains uniformly controlled. Thus, in this weighted Sobolev topology, \(O(ν^{2/3})\) is a sufficient stability scale without logarithmic loss.
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